## Section: Scientific Foundations

### Proof search in Type Theory

Cross-fertilizing ideas between the proof search approach and the
proof normalization approach, Lengrand has interacted with the TypiCal
(INRIA Saclay) and the r^{2} (INRIA Rocquencourt) project-teams.

In proof assistants based on the proof normalization approach, or Type
Theory, it is a hard challenge to design and understand their proof
search mechanisms. Based on ideas fromĀ [49] , a
major effort has been spent on using concepts from the proof search
approach, like *focused proof systems* , in order to rationalize
the implemented mechanisms.

By doing so, we have helped improve the Coq system, by impacting the
design of the new version of the tool's proof engine. One of these
proof search mechanisms, known as *pattern unification* , has again
become a hot topic of Coq's design, after Lengrand's use of Coq to
specify a particular algorithm has revealed a drastic need for this
missing feature.

It also emerged from Lengrand's interaction with these project-teams, that bridging Type Theory with the proof theory developed at Parsifal confirms the need for more extensionality on the functions programmed in Coq. Efforts to add such extensionality are ongoing.